= Transverse Doppler shift and observation time
{title2=$\lambda=\gamma\lambda_0(1-\beta\cos\theta)$}
Let $\theta$ be the angle between the source velocity and the emitted photon's propagation direction in the observer's frame. The <relativistic Doppler effect> gives $\nu=\nu_0/[\gamma(1-\beta\cos\theta)]$. A ray transverse in that frame has $\lambda=\gamma\lambda_0$, a redshift. A ray transverse in the source frame instead has $\cos\theta=\beta$ by <aberration of light>, and $\lambda=\lambda_0/\gamma$, a blueshift.
The time at which a passing source is closest must also be specified. Put the observer at the origin and the source at $(vt_e,d)$ at emission, with finite transverse offset $d>0$. Reception occurs at
$$
t_r=t_e+\frac{\sqrt{v^2t_e^2+d^2}}c.
$$
Emission at closest approach means $t_e=0$, a transverse observer-frame ray received at $t_r=d/c$, hence a redshift. Reception at closest approach means $t_r=0$, so $t_e=-\gamma d/c$ and the ray has $\cos\theta=\beta$, hence a blueshift. Sending $d$ to zero does not remove the distinction between these two limiting event prescriptions. https://web.mit.edu/8.286/www/quiz18/q1rp-euf18.pdf[MIT 8.286, transverse Doppler shifts, distinction between closest approach at emission and reception]{ref}
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